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Structure of exotic nuclei

Structure of exotic nuclei. Takaharu Otsuka University of Tokyo / RIKEN / MSU. A presentation supported by the JSPS Core-to-Core Program 　“ International Research Network for Exotic Femto Systems (EFES)”. 7 th CNS-EFES summer school Wako, Japan August 26 – September 1, 2008. Outline.

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Structure of exotic nuclei

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  1. Structure of exotic nuclei Takaharu OtsukaUniversity of Tokyo / RIKEN / MSU A presentation supported by the JSPS Core-to-Core Program　“International Research Network for Exotic Femto Systems (EFES)” 7th CNS-EFES summer school Wako, Japan August 26 – September 1, 2008

  2. Outline Section 1: Basics of shell model Section 2: Construction of effective interaction and an example in the pf shell Section 3: Does the gap change ? - N=20 problem - Section 4: Force behind Section 5: Is two-body force enough ? Section 6: More perspectives on exotic nuclei

  3. 2-body interaction Proton Neutron Aim: To construct many-body systems from basic ingredients such as nucleons and nuclear forces (nucleon-nucleon interactions) 3-body intearction

  4. Introduction to the shell model What is the shell model ? Why can it be useful ? How can we make it run ?

  5. Schematic picture of nucleon- nucleon (NN) potential Potential hard core 1 fm distance between nucleons 0.5 fm -100 MeV

  6. Actual potential Depends on quantum numbers of the 2-nucleon system (Spin S, total angular momentum J, Isospin T) Very different from Coulomb, for instance 1S0 Spin singlet (S=0) 2S+1=1 L = 0 (S) J = 0 From a book by R. Tamagaki (in Japanese)

  7. Basic properties of atomic nuclei Nuclear force = short range Among various components, the nucleus should be formed so as to make attractive ones (~ 1 fm )work. Strong repulsion for distance less than 0.5 fm Keeping a rather constant distance (~1 fm) between nucleons, the nucleus (at low energy)is formed.  constant density : saturation (of density)  clear surface despite a fully quantal system Deformation of surface Collective motion

  8. proton range of nuclear force from neutron Due to constant density, potential energy felt by is also constant Mean potential (effects from other nucleons) r Distance from the center of the nucleus -50 MeV

  9. proton range of nuclear force from neutron At the surface, potential energy felt by is weaker Mean potential (effects from other nucleons) r -50 MeV

  10. Eigenvalue problem of single-particle motion in a mean potential  Orbital motion Quantum number : orbital angular momentum l total angular momentum j number of nodes of radial wave function n E r Energy eigenvalues of orbital motion

  11. Neutron 中性子 Proton 陽子

  12. Mean potential Harmonic Oscillator (HO) potential HO is simpler, and can be treated analytically

  13. 5hw 4hw 3hw 2hw 1hw Eigenvalues of HO potential

  14. Spin-Orbit splitting by the (L S) potential An orbit with the orbital angular momentum l j = l - 1/2 j = l + 1/2

  15. magic number 20 2 8 Orbitals are grouped into shells shell gap closed shell fully occupied orbits The number of particles below a shell gap : magic number (魔法数) This structure of single-particle orbits shell structure (殻構造)

  16. 5hw 4hw 3hw 2hw 1hw Eigenvalues of HO potential Magic numbers Mayer and Jensen (1949) 126 82 50 28 20 8 2 Spin-orbit splitting

  17. From very basic nuclear physics, density saturation + short-range NN interaction + spin-orbit splitting Mayer-Jensen’s magic number with rather constant gaps Robust mechanism - no way out -

  18. Back to standard shell model How to carry out the calculation ?

  19. Hamiltonian ei :single particle energy vij,kl : two-body interaction matrix element ( i j k l : orbits)

  20. A nucleon does not stay in an orbit for ever. The interactionbetween nucleons changes their occupations as a result of scattering. Pattern of occupation : configuration 配位 mixing valence shell closed shell (core)

  21. How to get eigenvalues and eigenfunctions ? Prepare Slater determinantsf1, f2, f3 ,… which correspond to all possible configurations 配位 The closed shell (core) is treated as the vacuum. Its effects are assumed to be included in the single-particle energies and the effective interaction. Only valence particles are considered explicitly.

  22. < f1 | H | f2 >, < f1 | H | f1 >, < f1 | H | f3 >, .... aa+ ag+ ab+ f1 = ….. | 0 > aa’+ ag’+ ab’+ ….. | 0 > f2 = Step 1: Calculate matrix elements where f1 , f2 , f3 are Slater determinants In the second quantization, closed shell n valence particles f3 = ….

  23. < f1 |H| f3 > .... < f1 |H| f2 > < f1 |H| f1 > < f2 |H| f3 > .... < f2 |H| f2 > < f2 |H| f1 > < f3 |H| f1 > < f3 |H| f2 > < f3 |H| f3 > .... . . . < f4 |H| f1 > . . . . Step 2 : Construct matrix of Hamiltonian, H, and diagonalize it H=

  24. diagonalization Conventional Shell Model calculation All Slater determinants c diagonalization Quantum Monte Carlo Diagonalization method Important bases are selected (about 30 dimension) Diagonalization of Hamiltonian matrix

  25. Thus, we have solved the eigenvalue problem : H Y = E Y With Slater determinantsf1, f2, f3 ,…, the eigenfunction is expanded as Y = c1f1 + c2f2 + c3f3 + ….. ci probability amplitudes

  26. aa+ ag+ ab+ f1 = ….. | 0 > Usually single-particle state with good j, m (=jz) fi ’s has a good M (=Jz), Each of because M = m1 + m2 + m3 + ..... fi ’s having the same value of M are mixed. M-scheme calculation Hamiltonian conserves M. fi ’s But, having different values of M are not mixed.

  27. The Hamiltonian matrix is decomposed into sub matrices belonging to each value of M. M=0 M=1 M=-1 M=2 * * * * * * * * * * * * * * * * 0 0 0 H= * * * * * * * * * 0 0 0 * * * * * * * * * 0 0 0 . . . 0 0 0

  28. m1 m2 m1 m2 m1 m2 7/2 -3/2 5/2 -1/2 3/2 1/2 7/2 -7/2 5/2 -5/2 3/2 -3/2 1/2 -1/2 7/2 -5/2 5/2 -3/2 3/2 -1/2 J+ J+ How does J come in ? An exercise : two neutrons in f7/2 orbit J+ : angular momentum raising operator J+|j, m > |j, m+1 > M=2 M=0 M=1 J=1 can be elliminated, but is not contained J=0 2-body state is lost

  29. J = 0, 2, 4, 6 J = 2, 4, 6 J = 2, 4, 6 J = 4, 6 J = 4, 6 J = 6 J = 6 Dimension Components of J values M=0 4 M=1 3 M=2 3 2 M=3 M=4 2 M=5 1 1 M=6

  30. M = 0 eJ=0 0 0 0 0 eJ=2 0 0 0 0 eJ=4 0 0 0 0 eJ=6 * * * * * * * * * * * * * * * * H= By diagonalizing the matrix H, you get wave functions of good J values by superposing Slater determinants. In the case shown in the previous page, eJ means the eigenvalue with the angular momentum, J.

  31. M eJ 0 0 0 0 eJ’ 0 0 0 0 eJ’’ 0 0 0 0 eJ’’’ * * * * * * * * * * * * * * * * H= This property is a general one : valid for cases with more than 2 particles. By diagonalizing the matrix H, you get eigenvalues and wave functions. Good J values are obtained by superposing properly Slater determinants.

  32. Some remarks on the two-body matrix elements

  33. A two-body state is rewritten as | j1, j2, J, M > = Sm1, m2(j1, m1, j2, m2 |J, M) |j1, m1> |j2,m2> Clebsch-Gordon coef. Two-body matrix elements <j1, j2, J, M | V | j3, j4, J’, M’ > = Sm1, m2( j1, m1, j2, m2 |J, M) xSm3, m4( j3, m3, j4, m4 |J’, M’) x<j1, m1, j2, m2 | V | j3, m3, j4, m4 > Because the interaction V is a scalar with respect to the rotation, it cannot change J or M. Only J=J’ and M=M’ matrix elements can be non-zero.

  34. Two-body matrix elements <j1, j2, J, M | V | j3, j4, J, M > X X are independent of M value, also because V is a scalar. Two-body matrix elements are assigned by j1, j2, j3, j4 and J. Jargon : Two-Body Matrix Element = TBME Because of complexity of nuclear force, one can not express all TBME’s by a few empirical parameters.

  35. Actual potential Depends on quantum numbers of the 2-nucleon system (Spin S, total angular momentum J, Isospin T) Very different from Coulomb, for instance 1S0 Spin singlet (S=0) 2S+1=1 L = 0 (S) J = 0 From a book by R. Tamagaki (in Japanese)

  36. Determination of TBME’s Later in this lecture An example of TBME : USD interaction by Wildenthal & Brown sd shell d5/2, d3/2 and s1/2 63 matrix elemeents 3 single particle energies Note : TMBE’s depend on the isospin T Two-body matrix elements <j1, j2, J, T | V | j3, j4, J, T >

  37. USD interaction 1 = d3/2 2= d5/2 3= s1/2

  38. Effective interaction ～ Higher shell Excitations from lower shells are included effectively by perturbation(-like) methods Effects of core and higher shell valence shell Partially occupied Nucleons are moving around Closed shell Excitations to higher shells are included effectively

  39. 配位混合理論 Configuration Mixing Theory Departure from the independent-particle model Arima and Horie 1954 magnetic moment quadrupole moment This is included by renormalizing the interaction and effective charges. closed shell + Core polarization

  40. Probability that a nucleon is in the valence orbit ~60% A. Gade et al. Phys. Rev. Lett. 93, 042501 (2004) No problem ! Each nucleon carries correlations which are renormalized into effective interactions. On the other hand, this is a belief to a certain extent.

  41. In actual applications, the dimension of the vector space is a BIG problem ! It can be really big : thousands, millions, billions, trillions, .... pf-shell

  42. This property is a general one : valid for cases with more than 2 particles. By diagonalizing the matrix H, you get eigenvalues and wave functions. Good J values are obtained by superposing properly Slater determinants. M eJ 0 0 0 0 eJ’ 0 0 0 0 eJ’’ 0 0 0 0 eJ’’’ * * * * * * * * * * * * * * * * H= dimension Billions, trillions, … 4

  43. Dimension of shell-model calculations Dimension of Hamiltonian matrix (publication years of “pioneer” papers) Dimension billion Birth of shell model (Mayer and Jensen) Floating point operations per second Year Year

  44. Shell model code Name Contact person Remark OXBASH B.A. Brown Handy (Windows) ANTOINE E. Caurier Large calc. Parallel MSHELL T. Mizusaki Large calc. Parallel These two codes can handle up to 1 billion dimensions. (MCSM) Y. Utsuno/M. Honma not open Parallel

  45. Monte Carlo Shell Model Auxiliary-Field Monte Carlo (AFMC) method general method for quantum many-body problems For nuclear physics,Shell Model Monte Carlo (SMMC)calculation has been introduced by Koonin et al. Good for finite temperature. - minus-sign problem負符号問題 - only ground state, not for excited states in principle. Quantum Monte Carlo Diagonalization (QMCD) method No sign problem. Symmetriescan be restored. Excitedstates can be obtained.  Monte Carlo Shell Model 補助場（量子）モンテカルロ法

  46. References of MCSM method "Diagonalization of Hamiltonians for Many-body Systems by Auxiliary Field Quantum Monte Carlo Technique", M. Honma, T. Mizusaki and T. Otsuka, Phys. Rev. Lett. 75, 1284-1287 (1995). "Structure of the N=Z=28 Closed Shell Studied by Monte Carlo Shell Model Calculation", T. Otsuka, M. Honma and T. Mizusaki, Phys. Rev. Lett. 81, 1588-1591 (1998). “Monte Carlo shell model for atomic nuclei”, T. Otsuka, M. Honma, T. Mizusaki, N. Shimizu and Y. Utsuno, Prog. Part. Nucl. Phys. 47, 319-400 (2001)

  47. diagonalization Conventional Shell Model calculation All Slater determinants c diagonalization Quantum Monte Carlo Diagonalization method Important bases are selected (about 30 dimension) Diagonalization of Hamiltonian matrix

  48. Progress in shell-model calculations and computers Lines : 105 / 30 years Dimension of Hamiltonian matrix (publication years of “pioneer” papers) Monte Carlo Conventional Dimension More cpu time for heavier or more exotic nuclei 238U one eigenstate/day in good accuracy requires 1PFlops Year Birth of shell model (Mayer and Jensen) Floating point operations per second 京速計算機 (Japanese challenge) GFlops Blue Gene Earth Simulator Our parallel computer Year

  49. Outline Section 1: Basics of shell model Section 2: Construction of effective interaction and an example in the pf shell Section 3: Does the gap change ? - N=20 problem - Section 4: Force behind Section 5: Is two-body force enough ? Section 6: More perspectives on exotic nuclei

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